#200172

order	512
nilpotency class	3
isoclinism	isoclinic class of #9369
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

991 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,6,8,5,4,12,13)(3,9,10,16,11,7,14,15)(17,18,22,27,23,28,21,26)(19,29,33,40,32,39,31,24)(20,34,35,42,37,25,36,41)(30,43,44,38,46,48,45,47);(1,3,6,14,5,11,12,10)(2,7,13,15,4,9,8,16)(17,19,23,32)(18,24)(20,30,37,46)(21,31,22,33)(25,38)(26,39)(27,29)(28,40)(34,47)(35,44,36,45)(41,48)(42,43);(1,4)(2,5)(3,9)(6,8)(7,11)(10,15)(12,13)(14,16)(17,20)(18,25)(19,30)(21,35)(22,36)(23,37)(24,38)(26,41)(27,42)(28,34)(29,43)(31,44)(32,46)(33,45)(39,48)(40,47);(17,21,23,22)(18,26,28,27)(19,31,32,33)(20,35,37,36)(24,39,40,29)(25,41,34,42)(30,44,46,45)(38,48,47,43)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=hi,e^2=hi,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fhi,e^b=ei,f^b=fh,g^b=gi,e^c=eh,f^c=fi,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 107368);